2016edo
| ← 2015edo | 2016edo | 2017edo → |
2016 equal divisions of the octave (abbreviated 2016edo or 2016ed2), also called 2016-tone equal temperament (2016tet) or 2016 equal temperament (2016et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2016 equal parts of about 0.595 ¢ each. Each step represents a frequency ratio of 21/2016, or the 2016th root of 2.
Theory
2016edo shares the mapping for 3 with 224edo, albeit with a 28 relative cent error. First 7 prime harmonics with less than 25% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47.
2016edo has two reasonable mappings for 7. The 2016d val, ⟨2016 3195 4681 5659], tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and [14 11 -22 7⟩. As such, its circle of the interval 7/6 is the same as in 9edo. The patent val, ⟨2016 3195 4681 5658] tempers out 250047/250000, along with [7 18 -2 -11⟩ and [43 -1 -13 -4⟩. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo's patent val corresponds to 63/50.
In the 11-limit, 2016edo tempers out the [0 0 -22 0 3 11⟩ comma, which equates a stack of eleven 25/13's with three 11/1's. However, it does not temper out the jacobin comma.
2016 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked. One such temperament is 311 & 2016, produced by stacking 1465\2016, and defined for the 2.5.11.13.19.41 subgroup with the comma basis 16777475/16777216, 1171280/1171001, 615288025/615120896, 1180029296875/1179517976576.
Fractional-octave temperaments
The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of chromium with period 24 and the akjayland, period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18.
In the 2016dijk val, which is tuned better than the patent val, it supports the dike temperament, defined as 1600 & 2016dijk in the 37-limit with period 32.
In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as 1944 & 2016 and named after an eponymous song. It has a comma basis 47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.169 | -0.004 | +0.222 | +0.257 | -0.127 | -0.051 | -0.173 | -0.194 | +0.106 | +0.052 | +0.297 |
| Relative (%) | -28.4 | -0.7 | +37.2 | +43.1 | -21.4 | -8.6 | -29.1 | -32.5 | +17.8 | +8.8 | +49.9 | |
| Steps (reduced) |
3195 (1179) |
4681 (649) |
5660 (1628) |
6391 (343) |
6974 (926) |
7460 (1412) |
7876 (1828) |
8240 (176) |
8564 (500) |
8855 (791) |
9120 (1056) | |
Subsets and supersets
2016 is a significantly composite number, with its subset edos being 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008. Its abundancy index is 2.25. Some of its divisors have found applied use. 72edo has been used in Byzantine chanting, has been theoreticized by Alois Haba and Ivan Wyschnegradsky, and used by jazz musician Joe Maneri. 96edo has been used by Julian Carrillo, and 224edo is a member of zeta edos.
2016 is a divisor of some highly composite edos, such as 10080edo, 20160edo, etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160).
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-83 26 18⟩, [30 47 -45⟩ | [⟨2016 3195 4681]] | 0.036 | 0.050 | 8.4 |
| 2.3.5.7 | 250047/250000, [7 18 -2 -11⟩, [43 -1 -13 -4⟩ | [⟨2016 3195 4681 5660]] | 0.007 | 0.066 | 11.1 |
| 2.3.5.7 | 5250987/5242880, 40353607/40310784, [14 11 -22 7⟩ | [⟨2016 3195 4681 5659]] (2016d) | 0.060 | 0.060 | 10.1 |
| 2.3.5.11 | [14 8 -10 -1⟩, [-26 15 -5 4⟩, [-29 27 3 -6⟩ | [⟨2016 3195 4681 6974]] | 0.036 | 0.043 | 7.3 |
| 2.3.5.11.13 | 196625/196608, 53144100/53094899, [14 8 -10 -1 0⟩, [-13 9 5 -8 4⟩ | [⟨2016 3195 4681 6974 7460]] | 0.032 | 0.040 | 6.7 |
| 2.3.5.11.13.17 | 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608 | [⟨2016 3195 4681 6974 7460 8240]]] | 0.034 | 0.036 | 6.2 |
| 2.5.11.13.19.41.47 | 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865 | [⟨2016 4681 6974 7460 8564 10801 11198]] | 0.002 | 0.019 | 3.2 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 21 | 983\2016 (23\2016) |
585.119 (13.690) |
91875/65536 (126/125) |
Akjayland |
| 24 | 979\2016 (55\2016) |
582.738 (32.738) |
7/5 (?) |
Chromium |
| 32 | 29\2016 | 17.262 | (?) | Dike (2016dijk) |
| 72 | 925\2016 (1\2016) |
550.595 (0.595) |
73205/53248 (?) |
Jamala |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- Hemoclysm Totem (2022)